Voronoi Diagram

Voronoi diagrams are extremely versatile as a data structure for many geometric applications. Computing this diagram “exactly”, in particularly for a polyhedral set in 3-D, has been an unrealized quest of computational geometers for over two decades.

2D Voronoi Diagram

Overview

Here, we show an implementation of an alternative approach based on the well-known Subdivision Paradigm. Our unique emphasis is the use of purely numerical primitives. We avoid exact (algebraic) primitives because

they are hard to implement correctly, and

they fail to take full advantage of the resolution-limited properties of subdivision.

We encapsulate our numerical approach using the concept of soft primitives that conservatively converge to the exact ones in the limit.

Software

Software will be released soon.

Examples

A simple 2D example. The voronoi vertex is approximated as the center of the box containing it.
Same example, with finer subdivision.
This figure shows three line bisectors (in purple), one point bisector (in green) and two parabolas (in grey)
created by the features associated with the highlighted box (with red boundary).
A slightly more complex example
A polygone with hole. In this example, a Voronoi curve is created by connecting the mid points
of two box edges.